Subgroups — C₃×#90214 #13060868

Order 1344.

588 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6455	C₂	1	2	yes	maximal, complemented
5	C₃×D₇×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×(C₂²⋊D₇)⋊C₄	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×D₂₈	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×Dic₁₄	C₂	1	2	yes	maximal, complemented
9	#90214	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
12	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
13	C₃×#1157	C₄	1	8	yes	complemented
14	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₇	C₂²	1	4	yes	—
16	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
17	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₇	C₂²	1	2	yes	—
22	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
23	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
25	C₃×Dic₇⋊C₄	C₂²	1	4	yes	—
26	C₃×C₄×Dic₇	C₂²	1	4	yes	—
27	C₂×#1157	C₂×C₃	1	1	yes	characteristic
28	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
30	#6455	C₂×C₃	1	2	yes	complemented
31	D₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
32	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	4	yes	complemented
33	C₄×D₂₈	C₂×C₃	1	2	yes	complemented
34	C₄×Dic₁₄	C₂×C₃	1	2	yes	complemented
35	C₃×#520	—	7	1	no	maximal, Hall
36	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₃×D₇	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
39	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₃×C₂²⋊D₇	C₂×C₄	1	8	yes	complemented
44	C₂×C₃×Dic₇	—	4	2	no	—
45	C₂×C₃×Dic₇	C₂³	1	4	yes	—
46	C₂×C₃×Dic₇	C₂³	1	2	yes	—
47	C₃×D₂₈	C₂×C₄	1	4	yes	complemented
48	C₃×C₄×D₇	—	4	4	no	—
49	C₃×C₄×D₇	C₂×C₄	1	8	yes	complemented
50	C₃×Dic₁₄	C₂×C₄	1	4	yes	complemented
51	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
53	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
54	#1157	C₃×C₄	1	8	yes	complemented
55	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
57	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
58	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
59	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
62	C₄²×C₇	C₂²×C₃	1	2	yes	—
63	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
64	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
65	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
66	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
67	C₄×Dic₇	C₂²×C₃	1	4	yes	—
68	C₂×C₃×#41	—	7	1	no	—
69	C₃×C₄×Q₈	—	7	2	no	—
70	C₃×C₄×D₄	—	7	4	no	—
71	C₃×C₄×D₄	—	7	2	no	—
72	C₃×#117	—	7	2	no	—
73	C₃×#117	D₇	1	1	yes	characteristic, complemented
74	C₂×C₃×C₄⋊C₄	—	7	1	no	—
75	C₂×C₃×C₄⋊C₄	—	7	2	no	—
76	C₂²×C₃×C₇	—	2	1	no	—
77	C₂²×C₃×C₇	—	2	1	no	—
78	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₃×D₇	—	2	2	no	—
81	C₂×C₃×D₇	—	2	2	no	—
82	C₂×C₃×D₇	C₂²×C₄	1	4	yes	—
83	C₃×C₄×C₇	—	2	2	no	—
84	C₃×C₄×C₇	—	2	2	no	—
85	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
86	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₃×Dic₇	—	2	4	no	—
88	C₃×Dic₇	C₂²×C₄	1	4	yes	—
89	#520	—	7	1	no	Sylow
90	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
91	C₂²×D₇	C₂³×C₃	1	2	yes	—
92	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	complemented
93	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
94	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
95	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
97	C₂²⋊D₇	C₂×C₃×C₄	1	8	yes	complemented
98	C₂×Dic₇	—	4	2	no	—
99	C₂×Dic₇	C₂³×C₃	1	4	yes	—
100	C₂×Dic₇	C₂³×C₃	1	2	yes	—
101	D₂₈	C₂×C₃×C₄	1	4	yes	complemented
102	C₄×D₇	—	4	4	no	—
103	C₄×D₇	C₂×C₃×C₄	1	8	yes	complemented
104	Dic₁₄	C₂×C₃×C₄	1	4	yes	complemented
105	C₃×#41	—	7	8	no	—
106	C₂×C₃×Q₈	—	7	1	no	—
107	C₂×C₃×D₄	—	7	2	no	—
108	C₂×C₃×D₄	—	7	1	no	—
109	C₂²×C₃×C₄	—	7	2	no	—
110	C₂²×C₃×C₄	—	7	2	no	—
111	C₂²×C₃×C₄	—	7	4	no	—
112	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	7	4	no	—
114	C₃×C₄⋊C₄	—	7	2	no	—
115	C₃×C₄⋊C₄	—	7	2	no	—
116	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
117	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
118	C₃×C₂²⋊C₄	—	7	4	no	—
119	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
120	C₃×C₄²	—	7	4	no	—
121	C₃×C₄²	C₂×D₇	1	2	yes	—
122	C₂×C₃×C₇	—	2	2	no	—
123	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
126	C₃×D₇	—	2	4	no	—
127	C₂×#41	—	7	1	no	—
128	C₄×Q₈	—	7	2	no	—
129	C₄×D₄	—	7	4	no	—
130	C₄×D₄	—	7	2	no	—
131	#117	—	7	2	no	—
132	#117	C₃×D₇	1	1	yes	characteristic, complemented
133	C₂×C₄⋊C₄	—	7	1	no	—
134	C₂×C₄⋊C₄	—	7	2	no	—
135	C₂²×C₇	—	2	1	no	—
136	C₂²×C₇	—	2	1	no	—
137	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
138	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
139	C₂×D₇	—	2	2	no	—
140	C₂×D₇	—	2	2	no	—
141	C₂×D₇	C₂²×C₃×C₄	1	4	yes	—
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	2	no	—
144	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
145	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
146	Dic₇	—	2	4	no	—
147	Dic₇	C₂²×C₃×C₄	1	4	yes	—
148	C₂³×C₃	—	7	2	no	—
149	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
150	C₃×Q₈	—	7	4	no	—
151	C₃×D₄	—	7	8	no	—
152	C₃×D₄	—	7	4	no	—
153	C₂×C₃×C₄	—	7	2	no	—
154	C₂×C₃×C₄	—	7	8	no	—
155	C₂×C₃×C₄	—	7	4	no	—
156	C₂×C₃×C₄	—	28	4	no	—
157	C₂×C₃×C₄	—	28	2	no	—
158	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
159	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
160	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
161	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
163	C₃×C₇	#520	1	1	yes	characteristic, complemented, Hall
164	#41	—	7	8	no	—
165	C₂×Q₈	—	7	1	no	—
166	C₂×D₄	—	7	2	no	—
167	C₂×D₄	—	7	1	no	—
168	C₂²×C₄	—	7	2	no	—
169	C₂²×C₄	—	7	2	no	—
170	C₂²×C₄	—	7	4	no	—
171	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
172	C₄⋊C₄	—	7	4	no	—
173	C₄⋊C₄	—	7	2	no	—
174	C₄⋊C₄	—	7	2	no	—
175	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
177	C₂²⋊C₄	—	7	4	no	—
178	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
179	C₄²	—	7	4	no	—
180	C₄²	C₂×C₃×D₇	1	2	yes	—
181	C₂×C₇	—	2	2	no	—
182	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
183	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
185	D₇	—	2	4	no	—
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	7	4	no	—
189	C₂²×C₃	—	14	2	no	—
190	C₂²×C₃	—	14	2	no	—
191	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
192	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	7	4	no	—
196	C₃×C₄	—	14	4	no	—
197	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
198	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
199	C₂³	—	7	2	no	—
200	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
201	Q₈	—	7	4	no	—
202	D₄	—	7	8	no	—
203	D₄	—	7	4	no	—
204	C₂×C₄	—	7	2	no	—
205	C₂×C₄	—	7	8	no	—
206	C₂×C₄	—	7	4	no	—
207	C₂×C₄	—	28	4	no	—
208	C₂×C₄	—	28	2	no	—
209	C₂×C₄	C₃×C₄×D₇	1	4	yes	complemented
210	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
211	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
212	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
214	C₇	C₃×#520	1	1	yes	characteristic, complemented, Sylow
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	14	4	no	—
217	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
218	C₂×C₃	#6553	1	1	yes	characteristic
219	C₂×C₃	#6554	1	1	yes	characteristic
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	7	4	no	—
223	C₂²	—	14	2	no	—
224	C₂²	—	14	2	no	—
225	C₂²	C₂×C₃×C₄×D₇	1	2	yes	—
226	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	7	4	no	—
230	C₄	—	14	4	no	—
231	C₄	C₂×C₃×C₄×D₇	1	2	yes	—
232	C₄	C₂×C₃×C₄×D₇	1	2	yes	—
233	C₃	#90214	1	1	yes	characteristic, direct product factor, Sylow
234	C₂	—	2	2	no	—
235	C₂	—	14	4	no	—
236	C₂	C₂²×C₃×C₄×D₇	1	1	yes	characteristic
237	C₂	C₃×#6553	1	1	yes	characteristic
238	C₂	C₃×#6554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×#1157	C₂
cosocle	C₂²×C₇	C₂⁴×C₃
hypercenter	C₃×#117	D₇
hypocenter	C₇	C₃×#520
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×C₇×#117	C₂	semidirect
3. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
4. C₃×#6455	C₂	semidirect
5. C₃×D₇×C₄⋊C₄	C₂	semidirect
6. C₃×(C₂²⋊D₇)⋊C₄	C₂	semidirect
7. C₃×C₄×D₂₈	C₂	semidirect
8. C₃×C₄×Dic₁₄	C₂	semidirect
9. #90214	C₃	direct (N × H)
13. C₃×#1157	C₄	semidirect
28. C₇×#117	C₂×C₃	semidirect
29. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
30. #6455	C₂×C₃	semidirect
31. D₇×C₄⋊C₄	C₂×C₃	semidirect
32. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
33. C₄×D₂₈	C₂×C₃	semidirect
34. C₄×Dic₁₄	C₂×C₃	semidirect
38. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
43. C₃×C₂²⋊D₇	C₂×C₄	semidirect
47. C₃×D₂₈	C₂×C₄	semidirect
49. C₃×C₄×D₇	C₂×C₄	semidirect
50. C₃×Dic₁₄	C₂×C₄	semidirect
54. #1157	C₃×C₄	semidirect
73. C₃×#117	D₇	semidirect
92. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
97. C₂²⋊D₇	C₂×C₃×C₄	semidirect
101. D₂₈	C₂×C₃×C₄	semidirect
103. C₄×D₇	C₂×C₃×C₄	semidirect
104. Dic₁₄	C₂×C₃×C₄	semidirect
132. #117	C₃×D₇	semidirect
158. C₂×C₃×C₄	C₄×D₇	semidirect
163. C₃×C₇	#520	semidirect
209. C₂×C₄	C₃×C₄×D₇	semidirect
214. C₇	C₃×#520	semidirect
233. C₃	#90214	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#1157	2
C₃×C₇×#117	2
C₂×C₃×C₄⋊Dic₇	2
C₃×#6455	2
C₃×D₇×C₄⋊C₄	2
C₃×(C₂²⋊D₇)⋊C₄	2
C₃×C₄×D₂₈	2
C₃×C₄×Dic₁₄	2
#90214	3
C₃×#520	7

Subgroups — C3×#90214 #13060868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90214 #13060868